Question

The sides of two triangles are 35 cm, 53 cm, 66 cm and 56 cm, 33 cm, 65 cm respectively. Area of the circle is equal to the sum of the areas of these two triangles. Find the radius of the circle \[\left( {\sqrt 3 = 1.73} \right)\].
A) \[61.22cm\]
B) \[42.22cm\]
C) \[12.22cm\]
D) \[24.22cm\]

Answer 1

Hint:
Here we will first find the area of the triangle given by using the Heron's formula. Then we will add this area of the two triangles and equate it to the formula of the area of the circle. We will solve the equation to get the value of the radius of the circle.

Formula used:
We will use the following formulas:
1) Heron's formula: \[A\] is the area of the triangle i.e. \[A = \sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} \], \[s\] is the semi perimeter of the triangle and \[a\], \[b\] and \[c\] are the sides of the triangle.
2) Semi perimeter of the triangle\[s = \dfrac{{a + b + c}}{2}\], \[s\] is the semi perimeter of the triangle and \[a\], \[b\] and \[c\] are the sides of the triangle.

Complete step by step solution:
The sides of the first triangle are given as 35 cm, 53 cm, 66 cm.
First we will find the value of the semi perimeter of the triangle i.e. \[{s_1}\].
Substituting \[a = 35\], \[b = 53\] and \[c = 66\] in the formula \[s = \dfrac{{a + b + c}}{2}\], we get
\[{s_1} = \dfrac{{35 + 53 + 66}}{2}\]
Adding the terms in the numerator, we get
\[{s_1} = \dfrac{{154}}{2}\]
Dividing 154 by 2, we get
\[{s_1} = 77\]
So, we get the semi perimeter of the first triangle as 77cm.
Now by using the Heron's formula, we get
\[{A_1} = \sqrt {{s_1}\left( {{s_1} - a} \right)\left( {{s_1} - b} \right)\left( {{s_1} - c} \right)} = \sqrt {77\left( {77 - 35} \right)\left( {77 - 53} \right)\left( {77 - 66} \right)} = \sqrt {77\left( {42} \right)\left( {24} \right)\left( {11} \right)} = \sqrt {853776} = 924c{m^2}\]
Now we will find the area of the second triangle.
Given sides of the second triangle are56 cm, 33 cm, 65 cm
Now we will find the area of the second triangle by using the Heron's formula.
Firstly we will find the value of the semi perimeter of the triangle i.e. \[{s_2}\], we get
\[{s_2} = \dfrac{{a + b + c}}{2} = \dfrac{{56 + 33 + 65}}{2} = 77cm\]
Now by using the Heron's formula, we get
\[{A_2} = \sqrt {{s_2}\left( {{s_2} - a} \right)\left( {{s_2} - b} \right)\left( {{s_2} - c} \right)} = \sqrt {77\left( {77 - 56} \right)\left( {77 - 33} \right)\left( {77 - 65} \right)} = \sqrt {77\left( {21} \right)\left( {44} \right)\left( {12} \right)} = \sqrt {853776} = 924c{m^2}\]
It is given that the area of the circle is equal to the area of the sum of these two triangles. Therefore, we get
Area of the circle \[ = {A_1} + {A_2}\]
\[ \Rightarrow \pi {r^2} = {A_1} + {A_2} = 924 + 924 = 1848\]
Now we will solve this equation to get the value of the radius of the circle, we get
\[ \Rightarrow \dfrac{{22}}{7} \times {r^2} = 1848\]
\[ \Rightarrow {r^2} = 1848 \times \dfrac{7}{{22}} = 588\]
\[ \Rightarrow r = \sqrt {588} = 24.22cm\]
Hence, the radius of the circle is \[24.22cm\].

So, the option D is the correct option.

Note:
Here we should remember the basic Heron's formula for the calculation of the area of the triangle. We have to write down the units of the measurement along with their values. We should know that the area is the amount of surface covered by a shape in two dimensions. Area is generally measured in square units. We have to remember the formula of the area of the circle.
Area of the circle \[ = \pi {r^2} = \dfrac{\pi }{4}{d^2}\] where, \[r\] is the radius of the circle and \[d\] is the diameter of the circle.